I'm looking to clip a box out of a plane mesh so that only the intersect of both is shown, like in this picture.

Context: I have a plane-mesh that shows a video. on top of it I want to display the same video with a filter applied to it, clipped to a smaller region (square or circle) which can be moved by the mouse. like this

maybe someone knows of a better way to do this than the path I'm trying to go

You should use an image mask instead. For example, give your texture an alpha channel that is the shape of how you want to clip it, and copy it over via canvas.

Alternatively, if you are comfortable writing shaders, you can just write a shader that takes another image in as a mask and use that.

Finally, for actual geometric clipping (which honestly I don't think you need in this case) you can use something like clipper.js http://sourceforge.net/projects/jsclipper/ which admittedly is a bit gross to work with sometimes, but is really powerful.